Final answer:
To determine the volume of the solid enclosed by the given paraboloids, a triple integral in cylindrical coordinates is used with limits of integration based on the intersection of the paraboloids.
Step-by-step explanation:
To find the volume of the solid enclosed by the paraboloid y = x² + z² and y = 8 - x² - z², we can use a triple integral. The intersection of these paraboloids gives us the limits of integration. We can recognize that the region of interest is symmetric about the y-axis, which suggests using cylindrical coordinates (r, θ, y) may simplify our work.
Both paraboloids intersect where y = x² + z² = 8 - x² - z². Solving for y gives y = 4. This means our solid is bounded below by y = x² + z² and above by y = 8 - x² - z² with y ranging from 0 to 4. In cylindrical coordinates, this becomes y = r² and y = 8 - r², and r ranges from 0 to √4.
The volume integral in cylindrical coordinates is given by:
∫∫∫_V r dy dθ dr
Where V is the volume of the solid, and the limits for r are 0 to √4, for θ are 0 to 2π (due to symmetry about the y-axis), and for y are r² to 8 - r². Calculating this triple integral will yield the volume of the solid.